Are there disadvantages to teaching complex numbers as purely geometrical objects?

Question

Complex numbers are, or at least were to me, generally introduced like this:

There's no number whose square is negative. That's a shame! Well, whatever - we'll make one up! Set $i^2=-1$ and declare that all the usual rules of the real numbers apply.

This is a simple approach, and close to how the complex numbers were stumbled upon historically. But I don't think it gets to the heart of what complex numbers are, and less subjectively, I think it leads to confusion.

1. For one thing, if we're being rigorous, it's nonsense. What precisely does "all the usual rules apply" mean? What is your proof that a structure obeying the above rules even exists?
2. For me at least, it led to a feeling of unease, basically because I was intuitively aware of point (1).
3. It makes the geometrical facts about complex numbers seem very mysterious.
4. It leads to this type of confusion. In the words of 19th century mathematician John T. Graves: "I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. (...) If with your alchemy you can make three pounds of gold, why should you stop there?"

My personal approach would be something like:

If we take the plane with a polar coordinate system, and define the following fairly natural geometric operations on points, we find that they obey the same algebraic laws as multiplication and addition. We express this observation by calling these points complex numbers.

I would start by writing down $a+bi$ as $(a, b)$ before quickly defining $i=(0, 1)$, identifying the horizontal axis with $\mathbb R$ and switching to the conventional notation.

Is this a poor way to teach complex numbers to a student who is encountering them for the first time (such as a high school student)? Is it more difficult for students to understand than the conventional approach? I can see how it might be difficult to explain why the horizontal axis "is" $\mathbb R$. And what about students who go on to do math or science in college? Is there any reason why they might struggle?

Answer 1

I think there are serious pedagogical problems with such an approach. Here is a good general rule for explaining any kind of math:

Skipping over the motivation doesn't make something easier to understand.

As math educators, our instinct is always to simplify the math that we are presenting as much as possible. We always want students to understand the core idea first, and then build from that idea towards an understanding of the details. When presenting a new topic, we always try to strip away anything that seems extraneous or distracting.

But motivation isn't extraneous, and it isn't distracting. Motivation is necessary for human beings to understand mathematics. It's not part of the logical structure of mathematics, but it's crucial for developing intuition about the subject. If you don't know what something is for, or why it was invented, then you don't really have any hope of understanding it.

The narrative that negative numbers don't have square roots, so we should try making some up, is the motivation for complex numbers. It's where they came from. It's the only way for them to make sense. Absent this motivation, complex numbers will come across as completely arbitrary, and therefore incomprehensible. Students will feel like there's something they're missing, and they will be right.

You raise some objections to presenting this narrative, on the grounds that it leads to "confusion". I disagree with the spirit of these objections, for the following reason:

A student who asks questions is not necessarily "confused".

Often, students need to understand a subject pretty well to be able to ask intelligent questions about it. When a student asks how something they just learned can be generalized, or how to tell whether something really works, it means that they already understand it well enough to think about it critically. The goal of education is not to leave students dumbfounded.

Let me respond to your individual objections:

For one thing, if we're being rigorous, it's nonsense. What precisely does "all the usual rules apply" mean? What is your proof that a structure obeying the above rules even exists?

These are excellent questions, and I think any good explanation of complex numbers should address these directly, at least to some extent. The definition of complex numbers highlights the fact that we need to codify the rules of algebra (leading, ultimately, to abstract algebra), and raises the question of whether it suffices to have axioms for a mathematical object, or whether we really need to construct it.

Ultimately, this discussion is a bit too sophisticated for students at this level, but it should be possible to say something about it. A student who realizes that such issues arise in mathematics understands things much better than a student who thinks mathematics is always cut-and-dried.

It makes the geometrical facts about complex numbers seem very mysterious.

They are very mysterious. It's one of the great mysteries of mathematics that complex numbers work so well, that they are so intimately related to geometry, and that they arise in so many places. How did we start with square roots of negative numbers and arrive at Euler's formula? Why is complex analysis so much simpler and more beautiful than real analysis?

If something truly is mysterious, it's ok for students to perceive it that way. Point out explicitly how amazing and unexpected the geometry is, given that we didn't start with something geometric at all.

It leads to this type of confusion. In the words of 19th century mathematician John T. Graves: "I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. (...) If with your alchemy you can make three pounds of gold, why should you stop there?"

Again, this isn't confusion. The student who asked the question on Stack Exchange has grasped the fundamental technique of complex numbers, i.e. making up new number systems to do what we want, and wants to know how far it goes. Intellectually, they are doing as well as Graves, and they are on the conceptual path that eventually leads to abstract algebra.

A student who doesn't ask a question like this after learning about complex numbers hasn't really grasped them. If you manage to teach complex numbers in a way that doesn't prompt students to ask this question, that means that you've taught less, not more.

Of course, you won't have time to fully answer questions like this, but that's ok. Students at the high-school level sometimes get the feeling that mathematics is some kind of closed book, and this is a great opportunity to point out how much mathematics there really is in the world to explore. Graves' question is one of the most important questions in mathematics, the motivation for huge swaths of modern algebra, and in some sense it still hasn't been fully answered. It's not a question to avoid --- it's the kind of question you should go out of your way to mention!

Answer 2

It's funny you should ask this now, because I just taught a Math Ed graduate class on this topic the other day, so a lot of these thoughts are fresh in my mind.

The pedagogical sequence we conventionally follow with respect to complex numbers -- first introducing them in a "just pretend" way, and only later (for some students) providing a geometric interpretation/construction of them -- largely replicates the historical development of the subject. From the time complex numbers were first introduced in the 16th century, until Hamilton's reconstruction of the theory in the 19th century, "imaginaries" were generally regarded as a useful fiction, instrumental for the purpose of extracting real solutions to problems with real quantities in them, but intrinsically meaningless themselves. It was not until Hamilton introduced what we would now describe as a "vector" framework, showing that the field of complex numbers can be constructed out of real ingredients, that they began to escape their tainted reputation.

One drawback to the approach you suggest is that although you describe the multiplication operation as "fairly natural" -- which I agree it is when points/vectors are represented using polar coordinates -- there is a far more natural multiplication operation when points/vectors are represented in rectangular coordinates, namely $(a,b) \cdot (c,d) = (ac,bd)$, in precise analogy with how vector addition is defined (and I do not think you can plausibly argue that vector addition is best represented in polar coordinates). In contrast the "correct" definition $(a,b) \cdot (c,d) = (ac-bd,ad+bd)$ is harder to remember and seemingly unmotivated, except in hindsight. So this approach does away with the "let's pretend" quality of the complexes but at the cost of basing the theory on a seemingly arbitrary multiplication law.

Indeed, explicitly acknowledging this fact -- that there are at least two different ways one can define multiplication of vectors, one natural and one seemingly arbitrary -- can lead to some interesting investigations. How many different ways can multiplication on $\mathbb{R}^2$ be defined? What properties do those multiplication laws have? How many of them are fields? I still remember how exciting it was when I figured out that the only way to define multiplication on $\mathbb{R}^2$ that yields a field is, up to a change of basis, Hamilton's definition. It was like a missing piece of a jigsaw puzzle clicked into place. For me, that is the "motivation" for the complex numbers.

But insight like that can only come to people who already have familiarity with working with these numbers in a less formal way first. While I certainly do not think as educators we must be slaves to the historical development of a subject, I think there is probably some truth to the idea that Hamilton's insights would not have been possible if they had not been preceded by three centuries of people working formally with numbers whose true "meaning" even the most brilliant mathematicians did not really understand.

Edited, October 2020: I am resurrecting and updating this six-year old answer to respond to a request in the comments. Way back in May 2017, user "Andrew" asked:

Can you link to a proof that "the only way to define multiplication on $\mathbb R^2$ that yields a field is, up to a change of basis, Hamilton's definition."

I am happy to say that such a proof -- in fact three different proofs! -- have now been published in Chapter 6 of the textbook Secondary Mathematics for Mathematicians and Educators: A View from Above.

More specifically, in that chapter you will find the following theorem:

Theorem. There are infinitely many different ways to define a multiplication law on $\mathbb R^2$. However, up to a change of basis, each of these possible multiplicative structures is isomorphic to one of the following three possibilities:

1. $\mathbb C$, the field of complex numbers;
2. $\mathbb R \oplus \mathbb R$, the direct sum of two copies of the real number line;
3. $\mathbb R [\epsilon]$, the ring of dual numbers.

Of these three possibilities, only the first is a field.

This result is proved three different ways:

1. First, it is proved using nothing more complicated than very basic linear algebra (specifically, the ideas of a basis, and coordinates relative to a basis, and how to use determinants to tell whether a given system of 2 equations in two unknowns has a unique solution.
2. Next, it is proved again using the high school technique of "completing the square", and some clever change-of-variables tricks.
3. Third, it is proved using basic ideas of quotient rings, and the Chinese Remainder Theorem.

All of the above can be found on pp. 296-315 of the print edition of the book.

Answer 3

To answer the literal question of the title: yes, there are some disadvantages in portrayal of the complex numbers as "purely geometric" objects", although perhaps these disadvantages are more philosophical than mathematical. E.g., it is unclear whether or not we "have license" to declare that "the plane" is "numbers"... although we might have reasons to want to do so. That is, further, we might be unhappy that the arithmetic of complex numbers depends on two-dimensional geometry and geometric intuition for its sense and apparent self-consistency.

Historically, the appearance of complex numbers was taken more seriously in the solution of cubics with three real roots, but for which the formula irremediably involved complex numbers. (Taking square roots of negative numbers was a thing that could simply be prohibited or declared nonsensical.)

Hamilton's search for a way to give numerical sense to three-space led to the quaternions as non-commutative (division algebra) structure on four-space, and, in particular, showed the impossibility of putting such a structure on three-space. Graves found the octonions, and so on. All this work quite blatantly had the motivation of "identifying" geometry and numbers. Grassmann's work was similarly motivated.

But, yes, many people were uneasy. Kronecker's construction of $\mathbb C$ as a quotient $\mathbb R[x]/\langle x^2+1\rangle$ is not as elementary as one might wish, and not geometric, but is arguably a more bullet-proof argument for existence without appeal to geometric intuition. No, it doesn't explain why we cared in the first place, but it may give a skeptic more confidence.

As far as what beginning students should be told... probably not the Kronecker version, since this does not convey the intuitions that motivated things in the first place.

Answer 4

I think that including the geometry from the start is very important, as it is the geometry that explains why the complex numbers are useful and therefore why we bother teaching about them. After all, if it were all about the algebra then we'd continue with the quaternions, octonians, sontarans, and so forth. But we don't, because complex numbers find themselves used far more than those others and it is their interaction with geometry that is the reason for this.

I'm going to sketch out an approach that starts with the geometry and leads to the algebra. The key is to realise complex numbers as an orientation-preserving conformal linear transformations of the plane. Why should we be interested in such? Many reasons, but the basic idea is that these preserve the notion of "Turn Left" in that "turning left" commutes with such transformations[1].

Now, any such transformation consists of a rotation and a uniform scaling. The rotation is encoded by its angle and the scaling by its scale factor. So we need two numbers: $r$ and $\theta$. Composition of transformations results in $(r,\theta) (s,\phi) \mapsto (r s, \theta + \phi)$.

This is all very well, but it is more complicated than it need to be. Although the composition rule is nice, the application rule is not. That is, if we apply $(r,\theta)$ to $\begin{bmatrix}u \\ v \end{bmatrix}$ then the result is $\begin{bmatrix} r \cos(\theta) u - r \sin(\theta) v \\ r \sin(\theta) u + r \cos(\theta) v\end{bmatrix}$. So in terms of application, it is the numbers $x = r\cos(\theta)$ and $y = r\sin(\theta)$ that are important.

With that notation, the application rule becomes $\begin{bmatrix} u \\ v \end{bmatrix} \mapsto \begin{bmatrix} x u - y v \\ y u + x v \end{bmatrix}$.

This is also the composition rule, and that's because the vector $\begin{bmatrix} u \\ v \end{bmatrix}$ is the result of rotating the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ by a certain angle and scaling it by a certain length. So each vector in $\mathbb{R}^2$ corresponds to applying a rotation and scaling to $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. Thus we can identify $\mathbb{C}$onformal orientation-preserving linear transformations with $\mathbb{R}^2$ as the action is full and faithful.

Associativity of multiplication, aka composition, is obvious. Commutativity follows from the geometry in that rotations obviously commute with each other. Distributivity of multiplication over addition is also obvious since addition is translation and it's geometrically obvious how to change a "translate then rotate" (aka $z(w + c)$) in to "rotate then translate" (aka $z w + z c$).

As for the square root of minus one, well if you rotate by a quarter turn and rotate by a quarter turn again then you end up pointing backwards. Simple.

[1] I know that this shifts the burden of motivation from complex numbers to that of "knowing how to turn left", but that's hopefully easier to motivate directly from physics and engineering.

Answer 5

I think there are several significant disadvantages to such a presentation of complex numbers.

There's no number whose square is negative. That's a shame! Well, whatever - we'll make one up!

OK, don't do it like that though. What do you mean by "there's no number whose square is negative"? I could just as well say, "there's no number whose square is $2$ - well, I'll just make one up and call it $\sqrt 2$!"

The quote "God created the integers, all the rest is the work of man" is more or less true. All numbers, even starting with rationals, are (basically) fabrications of the imagination. They are helpful concepts that do not necessarily make any sense to our intuitions.

So in this sense, the statement "there's no number whose square is negative" is neither true nor false. It depends entirely on whether we have created such a number, and whether that creation has proved consistent and useful for doing mathematics.

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Might I say next that they are called "complex/imaginary numbers" for a reason. They are basically numbers, not geometrical objects. I don't think students are going to take very well to the idea that geometric objects can be the solutions to quadratic equations. They are used to numbers being the solutions to equations.

Also, the Argand Plane is, more than anything, a representation of the complex numbers - not the complex numbers themselves. The real number line, for that matter, is also a representation of the real numbers, but those numbers are not literally the points on that line.

Once you have introduced $i$, simply explain that complex numbers are traditionally written in the form $$a+bi$$ This expression has two real parameters, and therefore we can think of the set of complex numbers as a set of pairs of real numbers - in other words, the complex plane. Then point out that doing various operations to these "vectors" in the plane results in interesting things - for example, multiplying yields a new vector with angle the sum of the inputs' angles and length the product of the inputs' lengths.

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To summarize, in my opinion: you should certainly not exclude the "normal" presentation of $i$ when introducing it to your students, but you should also incorporate the geometric aspect of complex numbers to give the students another way of thinking about them - and if you feel that you can more effectively explain them from a geometric standpoint, then by all means spend more time on that! Teach what you're best at, but I wouldn't leave them entirely unprepared for the idea that complex numbers are actually "numbers" in some peoples' minds.

Answer 6

Disadvantages of Argand Plane as the main conceptual metaphor for ℂ are the same as of the “real line” for ℝ: both are insufficiently intuitive. Why some point of the line/plane should be labelled with “0”, not another? Euclidean lines and planes are translationally invariant, whereas ℝ and ℂ are not. Why should we choose some scale (i.e. distance between 0 and 1), not another one? Why this direction, not opposite, shall be “positive”? Euclidean metaphors for numbers are indeed very helpful, even in professional mathematics, but numbers appear only as relations between points, not points themselves. The thing ancient Pythagoreans had a good grasp of, but possibly ignored by some modern teachers.

Don’t know about English-speaking countries, but Ī̲ was taught about vectors in secondary school. On a line, if we fixed one non-zero vector (that is the difference between two distinct points), then multiplying it by any real number gives all vectors along the line. So, we actually need at least three points to represent a “real number” without applying arbitrary conventions. If we fix one point as “0”, another one as “1”, then for the variable point X the vector “0 X” divided by “0 1” gives an arbitrary real number. That’s how we arrive to the notorious “real line”.

The same about complex numbers: they are quotients of 2D Euclidean vectors just as real numbers are quotients of 1D Euclidean vectors (and quaternions are quotients of 3D Euclidean vectors, BTW). The only quirk with Argand Plane is orientation. Quotients of 2D Euclidean vectors form a field, but which order of catheti of 1:1:√2 triangle gives i and which gives −i? So we choose, by convention, one specific orientation (of two possible), similarly to choosing one of two roots of −1 for i, to make the metaphor perfect. Now complex numbers can serve in planar geometry and electrical engineering (where the natural direction of time helps to distinguish i from −i) as quotients of vectors. Or, if you like it, scale factors in similarity transformations. And if you label one point on the blackboard as “0” and another as “1”, then you are able to draw “the complex plane”, but its points only represent complex numbers with all aforementioned conventions. They are not numbers themselves, of course.

Formally, we can speak of ℂ itself as of some metric space, of which complex numbers are elements (i.e. points), and prove that it’s isometric to Euclidean plane. But this requires sufficient level of mathematical abstraction, beyond secondary education IMHO.

Answer 7

1. If you're concerned what we really mean by saying the "usual rules" apply, you really are concerned with the axioms of the real number system, and how they compare to the axioms of the complex number system. The answer is a bit complex but can be understood thoroughly by knowing the axiomatic definitions of fields.

2. Your "geometric" approach sounds good to me, but you omitted the part about complex numbers which makes them a field: the ability to multiply* any two complex numbers and meaningfully attain a new one. I grasped complex number multiplication by looking at graphic examples where you can see that it amounts to multiplying the magnitudes while adding the angles. And you can show that this new angular addition concept of multiplication still gives the correct answers for multiplying real numbers such as (1)(1)=1, (1)(-1)=-1, (-1)(-1)=1.

*EDIT: Or divide, except by 0. Thanks to Incnis Mrsi's comment!

Answer 8

I want to jump in because I see so many answers and comment debating about what is the correct primary meaning of complex numbers. If I where an electrical engineer, I would probably claim that complex numbers are really meant to describe impedance of electrical circuits - with a magnitude describing the resistance and an argument describing the phasing effect on periodic signals - and that the laws of addition and product are what is needed to correctly describe the impedance of several circuits in series or in parallel. That would be just as wrong as saying that the true meaning of complex number is in their geometrical interpretation, or in their algebraic construction, or in their use to compute antiderivatives of $x\mapsto \cos^{12}(x)$.

It is a given that there are several ways to approach complex numbers, and the fact that they appear consistently in so many ways is in my opinion the most important and amazing thing about them. But one cannot reach every such aspects and connexion in a course, and therefore I will conclude something obvious: choose an approach that fits the goals of the course and the future need of the students. Depending on this, you may choose one or several aspects, and some or all connections between them.

For example, you may construct complex numbers algebraically in chapter 1, to solve the issue of missing square roots; then in chapter 2 consider similarities of the plane and try to express them all in terms of their action on coordinates, observe that two parameters are sufficient and look at what happens when we compose too of them. You would then find out in chapter 3 that both points of view reconcile through complex numbers. That is something that is meaningful for undergraduates majoring in maths, to teach abstract construction and to show how connected maths are.

At a more advanced level, one can relate complex analysis and the geometry of conformal transformations.

At a more basic level, one can try to relate the algebraic meaning of complex numbers with trigonometry; etc.

Among the awesomeness that seems to have been overlooked in previous answers and comments, is the fact that adding roots to quadratic polynomials actually gives roots to all polynomials, even those with coefficients in the new field (that is far from true with $\mathbb{Q}$!)

One thing that seems clear to me, from a French perspective, is that the geometrical interpretation is usually overlooked, and thus feels foreign even to many advanced students. This is why I would try to put more geometry rather than less (it also fits my personal taste, to be honest). Unless you only have a very specific need for complex numbers as a tool in a certain context, in which case the angle of approach should be clearly dictated, teaching them insisting on only one single aspect is probably a wrong choice - irrespective of this aspect being algebraic or geometric.